I am aware that during the writing of The Hobbit, Tolkien must not have had much in mind for a large universe, as he was simply writing a children's book. Therefore, Thranduil was simply named 'the Elven-king'.

I'm not entirely sure whether or not he was mentioned as Thranduil in any of the LotR books (I have not read them), but I'm pretty sure he was mentioned by that name in the extended first film. If so, this leads me to believe that either Jackson created the name himself, or that the name was created before that (either by Tolkien or someone else), just not when the character was introduced.

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There was also a strange Elf clad in green and brown, Legolas, a
messenger from his father, Thranduil, the King of the Elves of
Northern Mirkwood. And seated a little apart was a tall man with a
fair and noble face, dark-haired and grey-eyed, proud and stern of
glance.

The Fellowship of the Ring, Book II, Chapter 2: The Council of Elrond.

Now of old the name of that forest was Greenwood the Great, and its
wide halls and aisles were the haunt of many beasts and of birds of
bright song; and there was the realm of King Thranduil under the oak
and the beech.

The Silmarillion, Of the Rings of Power and the Third Age.

Obviously the name is repeated in most of the Professor's works, but arguably The Lord of the Rings is the first published book to have it on it, and maybe (I'm not sure of this) it is firstly conceived in The Silmarillion.

As @Plutor comments, this doesn't prove that Thranduil is the same King of the Elves in the Hobbit. We can read in the quote below how Glóin makes a direct reference to the King of the Elves right after Legolas mentions the dungeons where they were kept by his father, thus confirming that Thranduil is the Elven-king featured in the Hobbit:

‘Not through lack of watchfulness,’ said Legolas; ‘but perhaps through
over-kindliness. [...] But Gandalf bade us hope still for his cure, and
we had not the heart to keep him ever in dungeons under the earth,
where he would fall back into his old black thoughts.’

‘You were less tender to me,’ said Glóin with a flash of his eyes, as
old memories were stirred of his imprisonment in the deep places of
the Elven-king’s halls.

‘Now come!’ said Gandalf. ‘Pray, do not
interrupt, my good Glóin. That was a regrettable misunderstanding,
long set right. If all the grievances that stand between Elves and
Dwarves are to be brought up here, we may as well abandon this
Council.’

The Fellowship of the Ring, Book II, Chapter 2: The Council of Elrond.